cp's OEIS Frontend

A124399 a(n) = 4^(n - bitcount(n)) where bitcount(n) = A000120(n).

%I A124399 #22 Jul 25 2023 04:04:11
%S A124399 1,1,4,4,64,64,256,256,16384,16384,65536,65536,1048576,1048576,
%T A124399 4194304,4194304,1073741824,1073741824,4294967296,4294967296,
%U A124399 68719476736,68719476736,274877906944,274877906944,17592186044416,17592186044416
%N A124399 a(n) = 4^(n - bitcount(n)) where bitcount(n) = A000120(n).
%C A124399 Numerators of one half of norm square of monic Legendre polynomials p_n(x).
%C A124399 The denominators of these polynomials are given by A069955.
%C A124399 The rationals N2(n) = 2*a(n)/A069955(n) give the minimal norm square for real monic polynomials. The norm square is defined as integral over the interval [-1,+1] of the square of the polynomials. Cf. the Courant-Hilbert reference.
%D A124399 Richard Courant and David Hilbert, Methoden der mathematischen Physik, Bd. I, 3, Auflage, Springer, 1993, pp. 73-74.
%H A124399 Wolfdieter Lang, <a href="/A124399/a124399.txt">Norm square, rationals and more</a>.
%F A124399 a(n) = numerator(N2(n)/2) with N2(n)/2:=(1/(2*n+1))*((2^n)/binomial(2*n,n))^2.
%F A124399 N2(n)/2 = (1/(2*n+1))*(1/L(n))^2 with L(n)= A001790(n)/A060818(n), the leading coefficient of the Legendre polynomial P_n(x), in lowest terms.
%F A124399 Bisection: a(2*n)=a(2*n+1) = A056982(n), n>=0.
%e A124399 Rationals a(n)/A069955(n): [1, 1/3, 4/45, 4/175, 64/11025, 64/43659, 256/693693, ...].
%e A124399 Rationals N2(n): [2, 2/3, 8/45, 8/175, 128/11025, 128/43659, 512/693693,...].
%t A124399 a[n_] := 4^(n - DigitCount[n, 2, 1]); Array[a, 25, 0] (* _Amiram Eldar_, Jul 25 2023 *)
%o A124399 (PARI) a(n) = numerator((1/(2*n+1))*((2^n)/binomial(2*n,n))^2); \\ _Michel Marcus_, Aug 11 2019
%o A124399 (Julia)
%o A124399 bitcount(n) = sum(digits(n, base=2))
%o A124399 a(n) = 4^(n - bitcount(n)) # _Peter Luschny_, Oct 01 2019
%Y A124399 Cf. A000120, A001790, A056982, A060818, A069955 (denominators of N2(n) as defined in the comments).
%K A124399 nonn,easy
%O A124399 0,3
%A A124399 _Wolfdieter Lang_, Nov 10 2006
%E A124399 New name by _Peter Luschny_, Oct 01 2019